Existence for a Cahn-Hilliard Model for Lithium-Ion Batteries with Exponential-Growth Boundary Conditions

The Cahn-Hilliard reaction model, a nonlinear, evolutionary PDE, was introduced to model phase separation in lithium-ion batteries. Using Butler-Volmer kinetics for electrochemical consistency, this model allows lithium-ions to enter the domain via a nonlinear Robin-type boundary condition $\partial_\nu \mu = R(c,\mu)$ for the chemical potential $\mu$, with $c$, the lithium-ion density. Importantly, $R$ depends exponentially on $\mu$. Fixed point methods are applied to obtain existence of regular solutions of the Cahn-Hilliard reaction model in dimension $3,$ allowing for recovery of exponential boundary conditions as in the physical application.